Having continued success performing a part on the radio (2,1,4)
I believe the answer is:
on a roll
'having continued success' is the definition.
I don't know anything about this answer so I cannot tell whether it can be defined by this definition.
'performing a part on the radio' is the wordplay.
'performing' becomes 'on' (as in being on stage).
'part' becomes 'role' (eg a theatrical role).
'on the radio' indicates a 'sounds like' (homophone) clue.
'role' is a homophone of 'roll'.
'on'+'a'+'roll'='ON A ROLL'
Can you help me to learn more?
(Other definitions for on a roll that I've seen before include "Where ham may be" , "Having a run of good luck (colloq.)" , "continually successful" , "Butter may be seen here" , "enjoying lots of success" .)